p-group, metabelian, nilpotent (class 2), monomial
Aliases: C43.3C2, C8⋊C4⋊12C4, (C2×C4).41C42, (C2×C42).31C4, C2.9(C4×M4(2)), C42.243(C2×C4), (C2×C4).58M4(2), C2.4(C42⋊4C4), C22.47(C2×C42), C4.68(C42⋊C2), C2.1(C42.6C4), (C22×C8).374C22, C23.248(C22×C4), (C2×C42).985C22, C22.35(C2×M4(2)), (C22×C4).1601C23, C22.46(C42⋊C2), C22.7C42.38C2, (C2×C8).128(C2×C4), (C2×C8⋊C4).21C2, (C2×C4).911(C4○D4), (C22×C4).432(C2×C4), (C2×C4).591(C22×C4), SmallGroup(128,477)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C43.C2
G = < a,b,c,d | a4=b4=c4=1, d2=c, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=bc2, cd=dc >
Subgroups: 188 in 136 conjugacy classes, 84 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C8⋊C4, C2×C42, C2×C42, C22×C8, C22.7C42, C43, C2×C8⋊C4, C43.C2
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, C4○D4, C2×C42, C42⋊C2, C2×M4(2), C42⋊4C4, C4×M4(2), C42.6C4, C43.C2
►Regular action on 128 points
Generators in S128
(1 119 103 39)(2 64 104 24)(3 113 97 33)(4 58 98 18)(5 115 99 35)(6 60 100 20)(7 117 101 37)(8 62 102 22)(9 96 128 56)(10 25 121 105)(11 90 122 50)(12 27 123 107)(13 92 124 52)(14 29 125 109)(15 94 126 54)(16 31 127 111)(17 81 57 41)(19 83 59 43)(21 85 61 45)(23 87 63 47)(26 72 106 73)(28 66 108 75)(30 68 110 77)(32 70 112 79)(34 82 114 42)(36 84 116 44)(38 86 118 46)(40 88 120 48)(49 80 89 71)(51 74 91 65)(53 76 93 67)(55 78 95 69)
(1 95 87 31)(2 92 88 28)(3 89 81 25)(4 94 82 30)(5 91 83 27)(6 96 84 32)(7 93 85 29)(8 90 86 26)(9 36 79 20)(10 33 80 17)(11 38 73 22)(12 35 74 19)(13 40 75 24)(14 37 76 21)(15 34 77 18)(16 39 78 23)(41 105 97 49)(42 110 98 54)(43 107 99 51)(44 112 100 56)(45 109 101 53)(46 106 102 50)(47 111 103 55)(48 108 104 52)(57 121 113 71)(58 126 114 68)(59 123 115 65)(60 128 116 70)(61 125 117 67)(62 122 118 72)(63 127 119 69)(64 124 120 66)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
G:=sub<Sym(128)| (1,119,103,39)(2,64,104,24)(3,113,97,33)(4,58,98,18)(5,115,99,35)(6,60,100,20)(7,117,101,37)(8,62,102,22)(9,96,128,56)(10,25,121,105)(11,90,122,50)(12,27,123,107)(13,92,124,52)(14,29,125,109)(15,94,126,54)(16,31,127,111)(17,81,57,41)(19,83,59,43)(21,85,61,45)(23,87,63,47)(26,72,106,73)(28,66,108,75)(30,68,110,77)(32,70,112,79)(34,82,114,42)(36,84,116,44)(38,86,118,46)(40,88,120,48)(49,80,89,71)(51,74,91,65)(53,76,93,67)(55,78,95,69), (1,95,87,31)(2,92,88,28)(3,89,81,25)(4,94,82,30)(5,91,83,27)(6,96,84,32)(7,93,85,29)(8,90,86,26)(9,36,79,20)(10,33,80,17)(11,38,73,22)(12,35,74,19)(13,40,75,24)(14,37,76,21)(15,34,77,18)(16,39,78,23)(41,105,97,49)(42,110,98,54)(43,107,99,51)(44,112,100,56)(45,109,101,53)(46,106,102,50)(47,111,103,55)(48,108,104,52)(57,121,113,71)(58,126,114,68)(59,123,115,65)(60,128,116,70)(61,125,117,67)(62,122,118,72)(63,127,119,69)(64,124,120,66), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)>;
G:=Group( (1,119,103,39)(2,64,104,24)(3,113,97,33)(4,58,98,18)(5,115,99,35)(6,60,100,20)(7,117,101,37)(8,62,102,22)(9,96,128,56)(10,25,121,105)(11,90,122,50)(12,27,123,107)(13,92,124,52)(14,29,125,109)(15,94,126,54)(16,31,127,111)(17,81,57,41)(19,83,59,43)(21,85,61,45)(23,87,63,47)(26,72,106,73)(28,66,108,75)(30,68,110,77)(32,70,112,79)(34,82,114,42)(36,84,116,44)(38,86,118,46)(40,88,120,48)(49,80,89,71)(51,74,91,65)(53,76,93,67)(55,78,95,69), (1,95,87,31)(2,92,88,28)(3,89,81,25)(4,94,82,30)(5,91,83,27)(6,96,84,32)(7,93,85,29)(8,90,86,26)(9,36,79,20)(10,33,80,17)(11,38,73,22)(12,35,74,19)(13,40,75,24)(14,37,76,21)(15,34,77,18)(16,39,78,23)(41,105,97,49)(42,110,98,54)(43,107,99,51)(44,112,100,56)(45,109,101,53)(46,106,102,50)(47,111,103,55)(48,108,104,52)(57,121,113,71)(58,126,114,68)(59,123,115,65)(60,128,116,70)(61,125,117,67)(62,122,118,72)(63,127,119,69)(64,124,120,66), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128) );
G=PermutationGroup([[(1,119,103,39),(2,64,104,24),(3,113,97,33),(4,58,98,18),(5,115,99,35),(6,60,100,20),(7,117,101,37),(8,62,102,22),(9,96,128,56),(10,25,121,105),(11,90,122,50),(12,27,123,107),(13,92,124,52),(14,29,125,109),(15,94,126,54),(16,31,127,111),(17,81,57,41),(19,83,59,43),(21,85,61,45),(23,87,63,47),(26,72,106,73),(28,66,108,75),(30,68,110,77),(32,70,112,79),(34,82,114,42),(36,84,116,44),(38,86,118,46),(40,88,120,48),(49,80,89,71),(51,74,91,65),(53,76,93,67),(55,78,95,69)], [(1,95,87,31),(2,92,88,28),(3,89,81,25),(4,94,82,30),(5,91,83,27),(6,96,84,32),(7,93,85,29),(8,90,86,26),(9,36,79,20),(10,33,80,17),(11,38,73,22),(12,35,74,19),(13,40,75,24),(14,37,76,21),(15,34,77,18),(16,39,78,23),(41,105,97,49),(42,110,98,54),(43,107,99,51),(44,112,100,56),(45,109,101,53),(46,106,102,50),(47,111,103,55),(48,108,104,52),(57,121,113,71),(58,126,114,68),(59,123,115,65),(60,128,116,70),(61,125,117,67),(62,122,118,72),(63,127,119,69),(64,124,120,66)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4AF | 8A | ··· | 8P |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | M4(2) | C4○D4 |
| kernel | C43.C2 | C22.7C42 | C43 | C2×C8⋊C4 | C8⋊C4 | C2×C42 | C2×C4 | C2×C4 |
| # reps | 1 | 4 | 1 | 2 | 16 | 8 | 16 | 8 |
Matrix representation of C43.C2 ►in GL5(𝔽17)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 6 | 11 | 0 | 0 |
| 0 | 11 | 11 | 0 | 0 |
| 0 | 0 | 0 | 8 | 14 |
| 0 | 0 | 0 | 3 | 9 |
G:=sub<GL(5,GF(17))| [4,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,6,11,0,0,0,11,11,0,0,0,0,0,8,3,0,0,0,14,9] >;
C43.C2 in GAP, Magma, Sage, TeX
C_4^3.C_2
% in TeX
G:=Group("C4^3.C2"); // GroupNames label
G:=SmallGroup(128,477);
// by ID
G=gap.SmallGroup(128,477);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,58,172]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b*c^2,c*d=d*c>;
// generators/relations